LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Matrix Inequalities and LMIs

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Matrix Inequality[edit | edit source]

Definition-1

A Matrix Inequality, , in the variable is an expression of the form

,

where and ,

Linear Matrix Inequality[edit | edit source]

Definition-2

A Linear Matrix Inequality, , in the variable is an expression of the form

,

where and ,

Bilinear Matrix Inequality[edit | edit source]

Definition-3

A Bilinear Matrix Inequality (BMI), , in the variable is an expression of the form

where , and , ,

Example[edit | edit source]

Consider the matrices and , where . It is desired to find a symmetric matrix satisfying the inequality

where . The elements of are the design variables in this problem, and although equation is indeed an LMI in the matrix , it does not look like the LMI in definition 3. For simplicity, let us consider the case of so that each matrix is of dimension , and Writing the matrix in terms of a basis , yields


Note that the matrices are linearly independent and symmetric, thus forming a basis for the symmetric matrix . The matrix inequality in equation can be written as

Defining and yields

which now resembles the definition of LMI given in definition 2. Through out this wiki book, LMIs are typically written in the matrix form of equation rather than the scalar form of definition 2.

External Links[edit | edit source]